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The present invention relates generally, to positioning systems and more particularly to positioning systems employing alternating electromagnetic fields, as well as to apparatus for localization and tracking.
Various types of positioning systems which employ alternating electromagnetic fields are known. The following U.S. Patents and foreign patent documents are believed to represent the state of the art for positioning systems:
U.S. Pat. Nos. 4,054,881 and 4,314,251 to Raab; U.S. Pat. No. 4,622,644 to Hansen; U.S. Pat. No. 4,737,794 to Jones; U.S. Pat. Nos. 4,613,866, 4,945,305 and 4,849,692 to Blood, U.S. Pat. Nos. 4,017,858 and 4,298,874 and 4,742,356 to Kuipers; U.S. Pat. No. 5,168,222 to Volsin et al; U.S. Pat. No. 5,0170,172 to Weinstein; and U.S. Pat. No. 5,453,686 to Anderson; WO 94/04938 to Bladen; U.S. Pat. No. 5,953,683 to Hansen; U.S. Pat. No. 5,831,260 to Hansen; U.S. Pat. No. 5,767,960 to Orman; U.S. Pat. No. 5,767,669 to Hansen; U.S. Pat. No. 5,744,953 to Hansen; U.S. Pat. No. 5,742,394 to Hansen; U.S. Pat. No. 5,640,170 to Anderson; U.S. Pat. No. 5,600,330 to Blood; U.S. Pat. No. 5,307,072 to Jones; U.S. Pat. No. 4,945,305 to Blood; U.S. Pat. No. 4,710,708 to Rorden; U.S. Pat. No. 4,346,384 to Raab; U.S. Pat. No. 4,328,548 to Crow; U.S. Pat. No. 4,298,874 to Kuipers; U.S. Pat. No. 3,983,474 to Kuipers; U.S. Pat. No. 3,868,565 to Kuipers; 3,644,825 to Davis
U.S. Pat. No. 4,710,708 to Rorden describes a positioning system which employs only one magnetic coil.
Genetic algorithms are described in Genetic algorithms in search: optimization and machine learning, D. Goldberg, 1989; and
An introduction to genetic algorithms, Melanie Mitchell, 1996.
PLL technology is described in Phase locked loop: simulation and applications, by Roland E. Best, McGraw-Hill Book Company, ISBN 0070060517.
The theory of non-linear filtering and its applications are discussed in:
H. J. Kushner, xe2x80x9cApproximations to Optimal Nonlinear Filtersxe2x80x9d. IEEE Trans. A C., Vol. AC-12, No. 5, October 1967;
A. Gelb, J. F. Kaspar,. Jr., R. A. Nash, Jr., C. E. Price, and A. A. Southerland, Jr.,xe2x80x9cApplied Optimal Estimationxe2x80x9d, M.I.T. Press, Cambridge, Mass., 1974;
B. D. O. Anderson, and J. B. Moore, xe2x80x9cOptimal Filteringxe2x80x9d, Prentice-Hall, Englewood Cliffs, N.J., 1979;
A. H. Jazwinski, xe2x80x9cStochastic Processes and Filtering Theoryxe2x80x9d, Academic Press, New York 1971, and
M. S. Grewal, and A. P. Andrews, xe2x80x9cKalman Filteringxe2x80x9d, Prentice-Hall, Upper Saddle River, N.J., 1993,
The field equation law is discussed in:
J. D. Jackson, xe2x80x9cClassical Electrodynamicsxe2x80x9d, John Wiley and Sons, New York New York, 1975.
The application of Extended Kalman Filters (EKF) to tracking in the context of radar is discussed, for example, in U.S. Pat. Nos. 5,075,694, 4,179,696, 3,952,304 and 3,935,572. Other tracking systems are discussed, for example, in U.S. Pat. Nos. 5,095, 467 and 4,855,932.
The Kalman Filter is a standard tool for xe2x80x9cdata fusionxe2x80x9d of different sensors. In U.S. Pat. No. 5,416,712 GPS signals and dead reckoning are combined by a Kalman Filter, and where the gyro bias is also calibrated. In U.S. Pat. No. 5,645,077 automatic drift compensation is discussed,
Simulated annealing-based algorithms are described in:
B. Aarts and J. Korst, xe2x80x9cSimulated Annealing and Boltzman Machines: A Stochastic Ap1proach to Combinatorial Optimization and Neural Computingxe2x80x9d, John Wiley and Sons Ltd. (ISBN: 0471921467);
M. E. Johnson (Ed.) xe2x80x9cSimulated Annealing (Sa and Optimization: Modem Algorithms with VLSI, Optimal Design and Missile Defense Applications)xe2x80x9d, Amer. Sciences Pr. (ISBN: 0935950184); and
R. Azencott (Ed.), xe2x80x9cSimulated Annealing: Parallelization Techniquesxe2x80x9d, Wiley Interscience Series in Discrete Mathematics, John Wiley and Sons Ltd. (ISBN: 0471532312),
The disclosures of all publications mentioned in the specification and of the publications cited therein are hereby incorporated by reference,
The present invention provides improved apparatus and method for positioning and tracking objects, and also to a non-linear Kalman Filter tracker.
A genetic algorithm is typically employed for solving the position equation to obtain the position and orientation of the detector.
A particular advantage of a preferred embodiment of the present invention is conservation of bandwidth by cessation of operation of transmitters, which are not providing useful information.
Selective activation of the transmitters is preferably performed periodically. The period is preferably selected automatically by the system to match the expected pace of change in the location of the moving body and the estimated distance between transmitters.
There is thus provided in accordance with a preferred embodiment of the present invention a tracking and positioning system including a plurality of transmitters distributed within a working space, and at least one sensors attached to at least one moving object and operative to sense transmissions arriving from the plurality of transmitters, a dynamic transmission activator operative to track at least one position parameters of at least one of the sensors and to selectively activate and deactivate individual sets of at least one transmitters from among the plurality of transmitters, at least partly responsively to at least one position parameters of at least one of the sensors.
Further in accordance with a preferred embodiment of the present invention the dynamic transmission activator is operative to activate at least one individual transmitter from among the plurality of transmitters if and only if at least one sensor is within the operating range of the individual transmitter.
Still further in accordance with a preferred embodiment of the present invention the dynamic transmission activator is operative to deactivate at least one individual transmitter from among the plurality of transmitters if and only if all sensors are outside of the operating range of the individual transmitter.
There is also provided in accordance with yet another preferred embodiment of the present invention a tracking method for tracking a moving object whose initial position is substantially unknown, the method including the steps of using a genetic algorithm to initially position the moving object, and tracking the moving object using a Kalman filter tracking method.
Further in accordance with a preferred embodiment of the present invention, the tracking method also includes at least once repositioning the moving object, during tracking, using the genetic algorithm.
A block diagram of the disclosed system, for positioning and tracking objects, is shown in FIG. 1d. In accordance with a preferred embodiment of the present invention a system comprises of N transmitters, where Nxe2x89xa76, and at least one probe sensor which detects at least 6 electromagnetic signals, each characterized by its own frequency. The probe sensor typically comprises a single magnetic field detector that is connected to a digital signal processing circuit. The analog output of the magnetic signal detector is a voltage signal proportional to the superposition of the N magnetic field transmitters at the coordinates xi, yi, zi, xcex8i, xcfx86i, where the index i denotes the position of the magnetic coil i. It is a particular feature of a preferred embodiment of the present invention that the antenna coils need not be exactly mutually orthogonal and certainly need not be arranged such that the centers of the antenna coils coincide.
The analog signal is digitized and is introduced to an integrated digital signal processor block, as an input data. The digitized input data from one of the magnetic detectors is then used by the digital signal processor unit to compute the position and orientation coordinates of the magnetic detector. The output from the digital signal processor unit is then transferred to the Data Communication unit and then to the System Control Unit. The refresh rate of the output data is typically of the order of few times per second to a few hundred times per second.
The detector may comprise a one-axis antenna coil, as illustrated in FIG. 2, or may alternatively comprise any other suitable type of one-axis magnetic field detector, such as a Hall-effect detector or a solid state component e.g. a magneto-resistive detector or a magneto-diode or a magneto-transistor. The digital signal processor unit typically comprises three modules: a tracking and control module, an envelope detector module and a position determination unit. The tracking and control subsystem is operative to increase the precision of the position determinations by decreasing the dynamic range of the input signal to the AID converter.
The output of the tracking and control module is supplied to an envelope detector, which is operative to determine the received envelope amplitudes (magnitude and sign) C1, . . . , CN of the N magnetic signals received from the N RF transmitters. The tracking and control subsystem preferably comprises a Linear Predictive Coding (LPC) module. The envelope detector module typically comprises of N identical envelope detectors (EDs) working in parallel. Optionally, each of the ED modules comprises two sub-modules: a Phase Lock Loop (hereafter PLL), and a System Synchronization Unit, which is called during the operation of the ED module to define the absolute sign of the signal amplitude. Alternatively, each ED module comprises three sub-modules operating in parallel and another sub-module that is called when a system synchronization is needed. The three modules are: a Phase Lock Loop, a Non-coherent absolute value envelope-detector, and a Sign Detection Unit. A fourth sub-module, System synchronization unit, is then called to define the absolute sign of the signal amplitude.
The output of the envelope detector is supplied to the position determination unit which is operative, based on the signed amplitude values supplied by the envelope detector, to provide an output indication of the position of the magnetic field detector in the sensor.
The operation of the position determination unit is typically based on solving N analytic equations with N unknowns.
There is further provided in accordance with a preferred embodiment of the present invention a system for monitoring of the position of at least one portions of an object, the system including a plurality of transmitters operative to transmit alternating magnetic fields within a three-dimensional space, and at least one positioning sensors arranged to be fixed to at least one corresponding portions of the object whose positions it is sought to monitor, each of the at least one positioning sensors including a magnetic field receiver having at least one active axes and operative to receive at least one component, lying along the at least one active axes respectively, of the alternating magnetic fields, and at least one digital signal processors for storing at least one characteristic of the magnetic fields as transmitted by the plurality of transmitters and comparing the at least one characteristic to at least one characteristic of the magnetic fields as received by at least a corresponding one of the at least one positioning sensors and, accordingly, determining and providing an output indication of at least one position characteristic of at least one corresponding portions of the object.
Further in accordance with a preferred embodiment of the present invention at least one sensors comprise a single sensor arranged to be fixed to a single portion of the object whose position it is sought to monitor.
Still further in accordance with a preferred embodiment of the present invention the at least one position characteristic comprises at least one dimension of the spatial position of the object portion. Preferably the at least one position characteristic also includes at least one dimension of the angular position of the object portion.
Additionally in accordance with a preferred embodiment of the present invention the at least one sensors comprise a plurality of sensors arranged to be fixed to a corresponding plurality of portions of the object whose positions it is sought to monitor.
Moreover in accordance with a preferred embodiment of the present invention the magnetic field receiver has a single (detection) active axis and is operative to receive the component of the alternating magnetic fields lying along the single (detection) active axis.
Preferably the plurality of transmitters are operative to continuously transmit said alternating magnetic fields.
There is also provided in accordance with another preferred embodiment of the present invention a system for monitoring the position of at least one portions of an object in three-dimensional space having three axes, the system including at least six magnetic transmitters each having a center and each operative to transmit alternating magnetic fields within a three-dimensional space, a transmitter orientation maintainer operative to maintain at least three of the transmitters in orientations such that at least a component of the magnetic field of at least one of the transmitters falls within each of the 3 axes of the 3-dimensional space, and wherein less than all of the centers of the transmitters coincide, at least one positioning sensors arranged to be fixed to at least one corresponding portions of the object whose positions it is sought to monitor, each of the at least one positioning sensors comprising a magnetic field receiver receiving the alternating magnetic fields from the at least six transmitters, and at least one digital signal processor for storing at least one characteristic of the magnetic fields as transmitted by the plurality of at least six transmitters and comparing the at least one characteristic to at least one characteristic of the magnetic fields as received by at least a corresponding one of the at least one positioning sensors and, accordingly, determining at least one position characteristic of at least one object portion.
Further in accordance with a preferred embodiment of the present invention the at least one digital signal processor is provided integrally with a corresponding one of the at least one positioning sensors.
Additionally in accordance with a preferred embodiment of the present invention at least 3 of the transmitters are separate physical units such that the at least three transmitters can be positioned at any 3 user-selected locations.
There is also provided in accordance with yet another preferred embodiment of the present invention a system for monitoring the position of at least one portions of an object in three-dimensional space having three axes, the system including at least six magnetic transmitters each having an active axial direction and each operative to transmit alternating magnetic fields within a three-dimensional space, a transmitter orientation maintainer operative to maintain at least three of the transmitters in orientations such that at least a component of the magnetic field of at least one of the transmitters falls within each of the 3 axes of the 3-dimensional space, and wherein less than all of the transmitters"" active axial directions are mutually orthogonal, at least one positioning sensors arranged to be fixed to at least one corresponding portions of the object whose positions it is sought to monitor, each of said at least one positioning sensors comprising a magnetic field receiver receiving said alternating magnetic fields from the at least six transmitters, and at least one digital signal processor for storing at least one characteristic of the magnetic fields as transmitted by the plurality of at least six transmitters and comparing said at least one characteristic to at least one characteristic of the magnetic fields as received by at least a corresponding one of the at least one positioning sensors and, accordingly, determining at least one position characteristic of at least one object portion.
Further in accordance with a preferred embodiment of the present invention at least first and second transmitters from among the at least six transmitters transmit in different frequencies.
Preferably at least first and second transmitters from among the at least six transmitters transmit in different phases.
Additionally in accordance with a preferred embodiment of the present invention at least one of the at least one digital signal processors is operative to simultaneously process magnetic field characteristics arriving from more than one of the at least six transmitters.
Still further in accordance with a preferred embodiment of the present invention comprising an RF trigger which is operative to trigger all of the transmitters, thereby to synchronize the transmitters. Additionally or alternatively the RF trigger provides a timing signal to at least one of the at least one sensors. Preferably at least one of the sensors computes the absolute phase of the at least six transmitters, based on said timing signal.
There is also provided in accordance with a preferred embodiment of the present invention a method for monitoring of the position of at least one portions of an object, the method including affixing at least one positioning sensors to at least one corresponding portions of the object whose positions it is sought to monitor, the sensors being operative to receive alternating magnetic fields existing within a three dimensional space containing the object, comparing at least one characteristic of the magnetic fields as transmitted to at least one characteristic of the magnetic fields as received by the sensors, and using a result of the comparing step as an input to a genetic natural selection process for determining and providing an output indication of at least one position characteristic of at least one corresponding portions of the object.
Further in accordance with a preferred embodiment of the present invention at least one of the sensors comprises a Linear Predicted Coding control loop operative to increase the dynamic range of incoming signals. Preferably at least one of the sensors comprises a solid-state component. Additionally or alternatively at least one of the sensors comprises a control loop to improve the dynamic range of the signal intensity without the use of electronic components common in the art.
Further in accordance with a preferred embodiment of the present invention, at least one of the sensors comprises a PLL configuration whose output is relatively insensitive to its input amplitude.
Still further in accordance with a preferred embodiment of the present invention, at least one dynamic property of the PLL apparatus does not substantially depend on the input amplitude of the PLL apparatus.
Still further in accordance with a preferred embodiment of the present invention, the bandwidth of the PLL apparatus does not substantially depend on the input amplitude of the PLL apparatus.
Additionally in accordance with a preferred embodiment of the present invention, the relaxation time constant of the PLL apparatus does not substantially depend on the input amplitude of the PLL apparatus.
The dynamic properties of the PLL, specifically bandwidth, and its relaxation time constant typically do not depend on the input amplitude.
There is also provided in accordance with a preferred embodiment of the present invention a method for monitoring of the position of at least one portions of an object, the method including positioning a plurality of transmitters operative to transmit alternating magnetic fields within a three-dimensional space and affixing at least one positioning sensors arranged to be fixed to at least one corresponding portions of the object whose positions it is sought to monitor, each of the at least one positioning sensors comprising a magnetic field receiver having at least one active axes and operative to receive at least one component, lying along the at least one active axes respectively, of the alternating magnetic fields and storing at least one characteristic of the magnetic fields as transmitted by the plurality of transmitters and comparing the at least one characteristic to at least one characteristic of the magnetic fields as received by at least a corresponding one of the at least one positioning sensors and, accordingly, determining and providing an output indication of at least one position characteristic of at least one corresponding portions of the object, wherein the storing, comparing, determining and providing step is performed locally rather than remotely.
There is also provided in accordance with yet another preferred embodiment of the present invention a method for monitoring the position of at least one portions of an object in three-dimensional space having three axes, the method including positioning at least six magnetic transmitters each having a center and each operative to transmit alternating magnetic fields within a three-dimensional space, including maintaining at least three of the transmitters in orientations such that at least a component of the magnetic field of at least one of the transmitters falls within each of the 3 axes of the 3-dimensional space, and wherein less than all of the centers of the transmitters coincide, affixing at least one positioning sensor to at least one corresponding portions of the object whose positions it is sought to monitor, each of the at least one positioning sensors comprising a magnetic field receiver receiving said alternating magnetic fields from the at least six transmitters, and storing at least one characteristic of the magnetic fields as transmitted by the plurality of at least six transmitters and comparing the at least one characteristic to at least one characteristic of the magnetic fields as received by at least a corresponding one of the at least one positioning sensors and, accordingly, determining at least one position characteristic of at least one object portion.
There is also provided in accordance with another preferred embodiment of the present invention a method for monitoring the position of at least one portions of an object in three-dimensional space having three axes, the method including positioning at least six magnetic transmitters each having an active axial direction and each operative to transmit alternating magnetic fields within a three-dimensional space, including maintaining at least three of the transmitters in orientations such that at least a component of the magnetic field of at least one of the transmitters falls within each of the 3 axes of the 3-dimensional space, and wherein less than all of the transmitters"" active axial directions are mutually orthogonal, affixing at least one positioning sensors arranged to be fixed to at least one corresponding portions of the object whose positions it is sought to monitor, each of said at least one positioning sensors comprising a magnetic field receiver receiving the alternating magnetic fields from the at least six transmitters, and storing at least one characteristic of the magnetic fields as transmitted by the plurality of at least six transmitters and comparing said at least one characteristic to at least one characteristic of the magnetic fields as received by at least a corresponding one of the at least one positioning sensors and, accordingly, determining at least one position characteristic of at least one object portion.
There is thus further provided in accordance with yet another preferred embodiment of the present invention pose tracking apparatus operative to track the pose of a moving object based on magnetic flux measurements taken in the vicinity of the moving object, the pose tracking apparatus including a non-linear Kalman filter-based tracker operative to receive magnetic flux measurements performed in the vicinity of the moving object, to operate a non-linear Kalman-type filter on the measurements, thereby to generate information regarding the pose of the moving object, and a pose indicator operative to provide an output indication of the information regarding the pose of the moving object.
Further in accordance with a preferred embodiment of the present invention the non-linear tracker includes an EKF (extended Kalman filter).
Additionally in accordance with a preferred embodiment of the present invention, the non-linear filter operates on a state vector whose components include pose coordinates and first time-derivatives of the pose coordinates.
Further in accordance with a preferred embodiment of the present invention the pose coordinates include 3 spatial coordinates and 2 orientation coordinates.
Further in accordance with a preferred embodiment of the present invention the apparatus also includes a transmitter array, which may include less than six operative transmitters, inducing magnetic flux in the vicinity of the moving object.
Still further in accordance with a preferred embodiment of the present invention the non-linear tracker employs a field equations transformation from the pose of the moving object to the magnetic flux measurements taken in its vicinity.
Still further in accordance with a preferred embodiment of the present invention the step of employing the field equation transformation includes computing a function h of a state vector "xgr", as follows:       h    ⁢          (      ξ      )        =                    C        0                    R        3              ⁢          (                                    3            ⁢                          A              1                        ⁢                          A              2                                            R            2                          -                  A          3                    )      
where
Co is a coefficient,
R is the distance between a detector detecting the magnetic flux measurements and a transmitter within the transmitter array; and
A1=xcex4x sin(xcex8)cos(xcfx86s)+xcex4y sin(xcex8s)sin(xcfx86s)+xcex4z cos(xcex8s)
A2=xcex4x sin(xcex8d)cos(xcfx86d)+xcex4y sin(xcex8d)sin(xcfx86d)+xcex4z cos(xcex8d)
A3=sin(xcex8s)cos(xcfx86s)sin (xcex8d)cos(xcfx86d)+sin(xcex8s)sin(xcfx86s)sin(xcex8d)sin(xcfx86d)+cos(xcfx86s)cos(xcfx86d)
and wherein the pose of the detector is (xd, yd, zd, xcex8d, xcfx86d) and the pose of the transmitter is (xs, ys, zs, xcex8s, xcfx86s), and where xcex4x, xcex4y and xcex4z denote the distance between the x, y and z components, respectively, of the detector""s pose and the transmitter""s pose.
Additionally in accordance with a preferred embodiment of the present invention the non-linear tracker approximates an elliptic integral, at least when the moving object is close to a transmitter within the transmitter array, by computing first and second terms of a Taylor series representing the elliptic integral.
Additionally in accordance with a preferred embodiment of the present invention, the approximated elliptic integral includes a correction to the above mentioned A1 and A3,             A      1        ⇒                  A        1            ⁢              (                  1          -          δ                )                        A      3        ⇒                  A        3            ⁢              (                  1          -          η                )                        δ      xe2x80x2        =                  5        8            ⁢                        (                      ρ            R                    )                2            ⁢              (                              7            ⁢                                          A                1                2                                            R                2                                              -          3                )                  η    =                  9        8            ⁢                        (                      ρ            R                    )                2            ⁢              (                              5            ⁢                                          A                1                2                                            R                2                                              -          1                )            
and where xcfx81 is the radius of the transmitter.
Still further in accordance with a preferred embodiment of the present invention the orientation component of the pose of the moving object is represented by two angles, continuous over time xcex8xe2x80x2 and xcfx86xe2x80x2, whose relationship with conventional polar coordinates xcex8xe2x80x2 and xcfx86 is as follows:   θ  =      {                                                                      θ                xe2x80x2                                                                                      if                  ⁢                                      xe2x80x83                                    ⁢                                      mod                    ⁡                                          (                                              θ                        ,                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          π                                                                    )                                                                      ≤                π                                                                                        -                                  θ                  xe2x80x2                                                                                                      if                  ⁢                                      xe2x80x83                                    ⁢                                      mod                    ⁡                                          (                                              θ                        ,                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          π                                                                    )                                                                       greater than                 π                                                    ⁢                  
                ⁢        ϕ            =              {                                                            ϕ                xe2x80x2                                                                                      if                  ⁢                                      xe2x80x83                                    ⁢                  θ                                =                                  θ                  xe2x80x2                                                                                                                          ϕ                  xe2x80x2                                +                π                                                                                      if                  ⁢                                      xe2x80x83                                    ⁢                  θ                                =                                  -                                      θ                    xe2x80x2                                                                                          
Still further in accordance with a preferred embodiment of the present invention, in order to avoid singularity, a dynamic offset is described by the following transformation:
xcex8=cosxe2x88x921[cos(xcex8xe2x80x2)cos(xcfx86xe2x80x2)]
xcfx86=cosxe2x88x921[{square root over (cos2(xcex8)+sin2(xcfx86xe2x80x2)cos2(xcex8xe2x80x2))}]
where xcex8 and xcfx86 include the orientation component of the moving object""s pose after the dynamic offset, and xcex8 and xcfx86xe2x80x2 include the orientation component of the moving object""s pose before the dynamic offset,
Still further in accordance with a preferred embodiment of the present invention the non-linear filter employs the following matrices and operations:
"xgr"k(xe2x88x92)="PHgr""xgr"kxe2x88x921(+)
where k is a time index, "xgr"k(xe2x88x92) is a state vector predictor, "xgr"k(+) is a state vector corrector, and "PHgr" is a state transition matrix,
Pk(xe2x88x92)="PHgr"Pkxe2x88x921(+)"PHgr"T+Q
where P(xe2x88x92) is an estimate error covariance matrix predictor, P(+) is an estimate error covariance matrix corrector and Q is a process noise covariance matrix,       H    k    =                    ∂                  h          ⁢                      (                          ξ              _                        )                                      ∂                  ξ          _                      ⁢          "LeftBracketingBar"                        ξ          k                ⁢                  (          -          )                    
where h is a sensitivity function and "xgr" is a state vector,
xe2x80x83Kk=Pk(xe2x88x92)HkT[HkPk(xe2x88x92)HkT+Rk]xe2x88x921
where Rk is a measurement noise covariance matrix,
"xgr"k(+)="xgr"k(xe2x88x92)+Kk{xcex6kxe2x88x92h["xgr"k(xe2x88x92)]}
where xcex6 denotes the magnetic flux measurements taken in the vicinity of the moving object, and
Pk(+)=[Ixe2x88x92KkHk]Pk(xe2x88x92)
Still further in accordance with a preferred embodiment of the present invention the magnetic flux measurements may include less than six magnetic flux measurements in the vicinity of the moving object.
Additionally in accordance with a preferred embodiment of the present invention, the non-linear tracker is operative to time-vary a measurement-noise covariance matrix R and a process-noise covariance matrix Q.
Further in accordance with a preferred embodiment of the present invention the time-varying R and Q includes:             R      k      dec        =                  R                  k          -          1                dec            ⁢              ⅇ                                            T              k                        -                          T                              k                -                1                                                          τ            decay                                          R      k        =                  R        k        dec            +              R        inf                        Q      k      dec        =                  Q                  k          -          1                dec            ⁢              ⅇ                                            T              k                        -                          T                              k                -                1                                                          τ            decay                                          Q      k        =                  Q        k        dec            +              Q        inf            
Still further in accordance with a preferred embodiment of the present invention at least one of the transmitters inducing a magnetic flux sampled by the measurements is self-calibrated.
Additionally in accordance with a preferred embodiment of the present invention, the non-linear filter is operative to calibrate the location of each of the self-calibrating transmitters.
Further in accordance with a preferred embodiment of the present invention the non-linear filter is operative to calibrate the intensity of each of the self-calibrating transmitters.
Still further in accordance with a preferred embodiment of the present invention the non-linear filter is operative to calibrate the radius of each of the self-calibrating transmitters.
Additionally in accordance with a preferred embodiment of the present invention, the non-linear filter is operative to calibrate the orientation of each of the self-calibrating transmitters.
Further in accordance with a preferred embodiment of the present invention, the tracker uses a state vector whose components comprise characteristic the self-calibrated transmitters inducing a magnetic flux sampled by the measurements, and wherein at least one of the characteristics is self-calibrated.
Still further in accordance with a preferred embodiment of the present invention, the non-linear tracker employs a measurement noise matrix R and a process noise covariance matrix Q at least one of which is generated by an adaptive process.
Additionally in accordance with a preferred embodiment of the present invention the adaptive process for Q comprises       Q    k    scc    =                    (                  α          ⁢                      (                                                                                ξ                    k                    velocity                                    ⁢                                      (                    +                    )                                                  -                                                      ξ                                          k                      -                      1                                        velocity                                    ⁢                                      (                    +                    )                                                                                                T                  k                                -                                  T                                      k                    -                    1                                                                        )                          )            2        +                  (                  1          -          α                )            ⁢              Q                  k          -          1                scc            xe2x80x83Qk=Qkacc+Q0acc
"xgr"kvelocity(+)=estimate of {{dot over (x)}, {dot over (y)}, {dot over (z)}, {dot over (xcfx86)}, {dot over (xcex8)}}
Further in accordance with a preferred embodiment of the present invention the adaptive process for R comprises
Rks,s=xcex2{xcex6ksxe2x88x92hs["xgr"k(xe2x88x92)]}2+(1xe2x88x92xcex2)Rkxe2x88x921s,s
RkRks,s+R0s,s